Functional calculus and ∗-regularity of a class of Banach algebras II

Abstract

In this article, we define a natural Banach * -algebra L 1 Q (G; A) for a C * -dynamical system (A, G, α) which is slightly bigger than L 1 (G; A) (they are the same if A is finite-dimensional). We will show that this algebra is * -regular if G has polynomial growth. The main result in this article extends the two main results in [C.W. Leung, C.K. Ng, Functional calculus and * -regularity of a class of Banach algebras, Proc. Amer. Math. Soc., in press].

Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra

“…By [7, Prop. 1], see also [21,Prop. 1.3], we need to show that ρ(f ) ≤ π(f ) holds for all f ∈ A whenever π, ρ are * -representations of A satisfying kerπ ⊂ kerρ.…”

*-regularity of certain generalized group algebras

“…G is said to be * -regular if its group algebra L 1 (G) is * -regular, for example, all polynomial growth groups are * -regular [5,Satz 2]. The class of * -regular groups has been studied by several authors (see [1][2][3][4][5]8,11]). It is worthwhile mentioning the fact that * -regular groups must be amenable [5,Korollar].…”

“…It is known that L 1 (G, A; α) is * -regular when G is abelian [9]. In [11] the authors developed the Dixmier's functional calculus to show that L 1 (G, A; α) is * -regular if G is of polynomial growth and A is of finite dimension. For the case of compact groups, up to the author's knowledge, the * -regularity of L 1 (G, A; α) is still not known.…”